Study On Stabilization Of Nonlinear Switched Systems Based On Immersion-Invariance Principle

A nonlinear switched system is composed of several nonlinear modes(or sub-systems)and a switching rule.Under the guidance of the switching rule,the system switches from one mode to another.Such a system is of great significance both in theory development and engineering applications.Therefore,it has attracted many researchers'interest.In recent years,abundant research results on stability and stabilization of switched nonlinear systems have been reported.However,these results are mainlyfor switched nonlinear systems with special structures,for example,triangle struc-ture,upper triangular structure or cascade structure.At present,there is no unified and effective analysis and design method for general nonlinear switched systems.The nonlinear control method based on normal norm is a method to solve the sta-bilization problem for nonlinear systems.But,the problem of how to obtain the proper coordinate transformation for a general subsystem and how to design a suit-able switching law to solve the stabilization problem of switched nonlinear systems is very difficult.In addition,the method based on normal form and most of the other control methods in the current research are based on Lyapunov functions of plants.However,It is almost impossible to construct Lyapunov functions for many complex practical systems.In recent years,the Immersion and Invariance(I&I)approach proposed by Astolfi and Ortega is a kind of nonlinear control method,which does not require the Lyapunov function of the plant.But how to overcome the limitations of the method itself for a wider application range,and how to design a suitable switching law and thus extend this method to nonlinear switched systems case are really challenging.At present,research results on these problems are quite limited.Based on immersion and invariance principle,this dissertation focuses on the stabilization problem for nonlinear switched systems.The main contributions are as follows:1.Based on the normal form,the stabilization problem for a class of nonlinear switched systems under switching signals with average dwell time is investigated.First of all,this dissertation investigates the design of a feedback controller via non-linear normal form for nonlinear systems originated from the aero-engine control systems.By solving the partial differential equation,we successfully find the co-ordinate transformation,through which the plant is transformed into its nonlinear normal form.Then,based on the nonlinear normal form,a stabilizing state feed-back control law and the Lyapunov function are obtained using a constructive design method.In addition,taking into account the relationship between the change of ex-ternal parameters and the engine system model,a switching system is introduced to describe the engine control system with varying Mach number and the nonlin-ear normal form approach is further extended to design the switched controllers to stabilize the more general switched nonlinear systems with average dwell time.2.The problem of immersion and invariant stabilization for nonlinear switched systems under arbitrary switchings is studied.First,a sufficient condition for im-mersion and invariance stabilization of general affine nonlinear switched systems is obtained.Then,the relevant results are applied to the switched nonlinear system in strict feedback form and the immersion and invariance state feedback controllers for subsystems are constructed to globally asymptotically stabilize the closed-loop sys-tem under arbitrary switching.It is worth mentioning that the problem of immersion and stabilization for nonlinear switched systems is studied for the first time.The plant is not required to posses any special structure and the novel control method does not require the Lyapunov function of the plant.3.The problem of immersed and invariant stabilization for nonlinear switched systems with average dwell time is investigated.First of all,inspired by the aero-engine control problem,a new approach based on the integration of immersion and invariance(I&I)theory and viability theory-two relatively new tools,is presented,and a sufficient condition for stabilization of nonlinear systems with state constraints is drived.Then,the proposed method is applied to the system in strict feedback form.We design a state feedback controller constructively to ensure the immer-sion and invariance stabilizability of the original system,and constraints are not destroyed simultaneously.Secondly,a sufficient condition for immersion and invari-ance stabilization of switched nonlinear systems with average dwell time is given.The viability theory is further introduced to drive a sufficient condition for stabi-lization of nonlinear switched systems with state constraints.Then,the relevant results are applied to the switched system in strict feedback form with state con-straint represented by inequalities.It is worth pointing out that the introducing of viability theory provides another way to guarantee the trajectory boundedness of the closed-loop system and ensures that the constraints of the system are not destroyed.4.The problem of immersion and invariance adaptive stabilization for a class of linear parametric switched nonlinear systems is investigated,where the solvabil-ity of the immersion and invariance adaptive stabilization problem for subsystems is not assumed.First,an adaptive law with an additional nonlinear term is cho-sen to ensure the uniform stability of the error switching dynamics under arbitrary switchings.Furthermore,a state-dependent switching law is designed by using the generalized multiple Lyapunov function(GMLFs)method to guarantee the stabil-ity of the closed-loop system.Then,along with the Backstepping technique,the proposed control method is applied to linearly parameterized nonlinear switched systems in strict-feedback form.The common stabilizing functions are successfully found so as to avoid the use of different coordinate transformation.Also,the im-mersion and invariant adaptive state feedback controllers and error estimators for subsystems together with a state-dependent switching law are designed simultane-ously.